# L8a6

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a6 at Knotilus! L8a6 is ${\displaystyle 8_{6}^{2}}$ in the Rolfsen table of links.

 Planar diagram presentation X6172 X12,3,13,4 X16,8,5,7 X14,10,15,9 X10,14,11,13 X8,16,9,15 X2536 X4,11,1,12 Gauss code {1, -7, 2, -8}, {7, -1, 3, -6, 4, -5, 8, -2, 5, -4, 6, -3}
A Braid Representative

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {2uv-3u-3v+2}{{\sqrt {u}}{\sqrt {v}}}}}$ (db) Jones polynomial ${\displaystyle -{\frac {1}{q^{9/2}}}-q^{7/2}+{\frac {1}{q^{7/2}}}+2q^{5/2}-{\frac {3}{q^{5/2}}}-2q^{3/2}+{\frac {3}{q^{3/2}}}+3{\sqrt {q}}-{\frac {4}{\sqrt {q}}}}$ (db) Signature -1 (db) HOMFLY-PT polynomial ${\displaystyle a^{5}z^{-1}-2a^{3}z-a^{3}z^{-1}-za^{-3}+az^{3}+z^{3}a^{-1}+za^{-1}}$ (db) Kauffman polynomial ${\displaystyle -az^{7}-z^{7}a^{-1}-a^{2}z^{6}-2z^{6}a^{-2}-3z^{6}-a^{3}z^{5}+2az^{5}+2z^{5}a^{-1}-z^{5}a^{-3}-a^{4}z^{4}+7z^{4}a^{-2}+8z^{4}-a^{5}z^{3}-a^{3}z^{3}-3az^{3}+3z^{3}a^{-3}-5z^{2}a^{-2}-5z^{2}+2a^{5}z+2a^{3}z+az-za^{-3}+a^{4}-a^{5}z^{-1}-a^{3}z^{-1}}$ (db)

### Khovanov Homology

The coefficients of the monomials ${\displaystyle t^{r}q^{j}}$ are shown, along with their alternating sums ${\displaystyle \chi }$ (fixed ${\displaystyle j}$, alternation over ${\displaystyle r}$).
 \ r \ j \
-4-3-2-101234χ
8        11
6       1 -1
4      11 0
2     21  -1
0    21   1
-2   23    1
-4  11     0
-6  2      2
-811       0
-101        1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-2}$ ${\displaystyle i=0}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session.